[Maxima-discuss] Early references on undefined variables?

[Eduardo O. <edu...@gm...>]

Hi list, especially Richard,

I've read a part of Richard's PhD thesis,

  https://apps.dtic.mil/sti/pdfs/AD0740132.pdf

and it gave me the impression that this would be a good place to ask
this... so here it goes. But first a little story.

In the middle of my master's degree, ages ago, I switched from trying
to do research on Maths (that I was finding boring) to trying to do
research on Logic (that looked much more fun). The people in the Logic
group in my university were working mostly on Proof Theory, and every
time that they would start to work on a new logical system they would
try to prove strong normalization for it - and they would usually
succeed, because they had a lot of knowledge about which systems
"looked like something that they would be able to prove strong
normalization for"... systems that were not strongly normalizable were
"bad" to them, and they were put in the box of the toys that they
didn't want to play with. I somehow managed to 1) not learn how to do
strong normalization proofs, and to 2) focus more on Lambda Calculus
than on Proof Theory... so I know a lot of the terminology about
reductions and normalization, but only a few of the techniques.

End of the little story.

So: in Maxima we can change how our reductions work by changing lots
of flags, and in some contexts we can set, say, w to 42, and this
tells the system that from that point onwards every w should be
reduced to 42. And if we turn off most of the actions of the
simplifier we can make both of these expressions

  expr1 : (x + y)(x - y)
  expr2 : (x + y)(x - y) - (x^2 - y^2)

reduce to themselves.

I _guess_ that:

  1. when the first Lisps were being created people saw very quickly
     that evaluation should be different from β-reduction... in
     β-reduction and its variants a free variable like x reduces to
     itself, but it's better to define evaluation in a way in which
     evaluating an undefined variable yields an error... and:

  2. when the first Computer Algebra systems were being created people
     saw very quickly that variables should be treated in other, more
     complex, ways - both expr1 and expr2 above are valid expressions
     that can be manipulated in many ways, but that can't evaluated
     numerically or plotted because they both depend on both x and
     y... and I guess that even the first CASs had simple functions,
     that didn't work in all cases, that could compare expr1 and expr2
     to 0, and answer that "expr1 is not 0 because expr1 depends on x
     and y and 0 doesn't" - and the same for expr2 (ta-da).

My question is: can you recommend good early references that discuss
this, i.e., that discuss how the notion of free/undefined variables in
CASs was invented?

  Thanks in advance!
    Cheers =),
      Eduardo Ochs
      http://angg.twu.net/eev-maxima.html